Truncation and augmentation of level-independent QBD processes

In the study of quasi-birth-and-death (QBD) processes, the first passage probabilities from states in level one to the boundary level zero are of fundamental importance. These probabilities are organized into a matrix, usually denoted by G. The matrix G is the minimal nonnegative solution of a matrix quadratic equation. If the QBD process is recurrent, then G is stochastic. Otherwise, G is sub-stochastic and the matrix equation has a second solution Gsto, which is stochastic. In this paper, we give a physical interpretation of Gsto in terms of sequences of truncated and augmented QBD processes. As part of the proof of our main result, we derive expressions for the first passage probabilities that a QBD process will hit level k before level zero and vice versa, which are of interest in their own right. The paper concludes with a discussion of the stability of a recursion naturally associated with the matrix equation which defines G and Gsto. In particular, we show that G is a stable equilibrium point of the recursion while Gsto is an unstable equilibrium point if it is different from G.